Best Known (26−6, 26, s)-Nets in Base 5
(26−6, 26, 5211)-Net over F5 — Constructive and digital
Digital (20, 26, 5211)-net over F5, using
- net defined by OOA [i] based on linear OOA(526, 5211, F5, 6, 6) (dual of [(5211, 6), 31240, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(526, 15633, F5, 6) (dual of [15633, 15607, 7]-code), using
- construction X4 applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(525, 15625, F5, 6) (dual of [15625, 15600, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(519, 15625, F5, 4) (dual of [15625, 15606, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(57, 8, F5, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,5)), using
- dual of repetition code with length 8 [i]
- linear OA(51, 8, F5, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(5) ⊂ Ce(3) [i] based on
- OA 3-folding and stacking [i] based on linear OA(526, 15633, F5, 6) (dual of [15633, 15607, 7]-code), using
(26−6, 26, 12927)-Net over F5 — Digital
Digital (20, 26, 12927)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(526, 12927, F5, 6) (dual of [12927, 12901, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(526, 15632, F5, 6) (dual of [15632, 15606, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(525, 15625, F5, 6) (dual of [15625, 15600, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(519, 15625, F5, 4) (dual of [15625, 15606, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(51, 7, F5, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- discarding factors / shortening the dual code based on linear OA(526, 15632, F5, 6) (dual of [15632, 15606, 7]-code), using
(26−6, 26, 518874)-Net in Base 5 — Upper bound on s
There is no (20, 26, 518875)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 1 490120 388459 790501 > 526 [i]